Pattern formation in a predator–prey model on network and non-network environments

Pattern formation in mathematical modeling of biology has been a popular topic for a long time. In this study, we focus on possible patterns driven by Turing instability under the prey–predator model of nonlinear reaction–diffusion type in 2D space with the Allee effect and intra-species competition. We investigate pattern formation in predator–prey systems on both continuous media and network frameworks, highlighting how diffusion in continuous space and its Laplacian analogue on networks shape the emergence of Turing patterns. We derive the model from scratch, discuss three separate cases, and analyze the dynamics of both temporal and spatiotemporal dynamics of the models. Under circumstances where the Turing pattern does not exist, we present detailed proofs and use numerical simulation to demonstrate temporal Hopf bifurcation and patchy invasion. On the other hand, if the analytical conditions for the Turing pattern can be satisfied, we display patterns in our numerical simulations and give corresponding biological interpretations at the end.